Testing for a Threshold in Models with Endogenous Regressors

Using 2SLS estimation, we propose two tests for a threshold in models with
endogenous regressors: a sup LR test and a sup Wald test. Here, the 2SLS estimation is not conventional because it uses additional information about the first-stage being linear or not. Because of this additional information, our tests can be more accurate than the threshold test in Caner and Hansen (2004) which is based on conventional GMM estimation.
We derive the asymptotic distributions of the two tests for a linear and for a threshold reduced form. In both cases, the distributions are non-pivotal, and we propose obtaining critical values via a fixed regressor wild bootstrap.
Our simulations show that in small samples, the GMM test of Caner and Hansen (2004) can be severely oversized under heteroskedasticity, while the 2SLS tests we propose are much closer to their nominal size.
We use our tests to investigate the common claim that the government spending multiplier is larger close to the zero lower bound, and therefore that the governments should have spent more in the recent crisis. We find no empirical support for this claim.